1. Basic Terminology
Right Triangle Terminology:
• Line of Sight: Line from eye to object
• Angle of Elevation: Angle above horizontal
• Angle of Depression: Angle below horizontal
• Horizontal Line: Level line through observer's eye
2. Angle of Elevation
When looking at an object ABOVE the horizontal level:
Angle of Elevation = ?
tan ? = Height of object / Distance from object
Formula:
Height = Distance × tan(angle of elevation)
Distance = Height / tan(angle of elevation)
3. Angle of Depression
When looking at an object BELOW the horizontal level:
Angle of Depression = ?
tan ? = Depth / Distance from object
Important: Angle of elevation from A to B = Angle of depression from B to A
4. Basic Trigonometric Formulas for Height and Distance
Using tan ?
Height = Distance × tan ?
Distance = Height / tan ?
Using sin ?
Height = Hypotenuse × sin ?
Hypotenuse = Height / sin ?
Using cos ?
Distance = Hypotenuse × cos ?
Hypotenuse = Distance / cos ?
5. Common Problem Types
- Single Object Problems:
- Finding height of tower/building
- Finding distance from object
- Finding angle of elevation/depression
- Two Object Problems:
- Two observers, one object
- One observer, two objects
- Finding distance between two objects
- Moving Object Problems:
- Object moving towards/away from observer
- Changing angles of elevation
6. Step-by-Step Approach
- Draw a clear diagram
- Mark known values (height, distance, angles)
- Identify right triangles
- Choose appropriate trigonometric ratio
- Set up equation
- Solve for unknown
- Include units in final answer
7. Important Trigonometric Values
tan 30° = 1/v3 ˜ 0.577
tan 45° = 1
tan 60° = v3 ˜ 1.732
sin 30° = 1/2, sin 45° = 1/v2, sin 60° = v3/2
cos 30° = v3/2, cos 45° = 1/v2, cos 60° = 1/2
8. Shadow Problems
When sun's rays make angle ? with ground:
Length of shadow = Height of object / tan ?
Height of object = Length of shadow × tan ?
9. Quick Reference Formulas
Height Calculation
h = d × tan ?
Distance Calculation
d = h / tan ?
Using Pythagoras
AC² = AB² + BC²
10. Example Problems (NCERT Style)
Problem 1: A tower stands vertically on the ground. From a point on the ground which is 15m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. Find the height of the tower.
Solution:
Let height = h
tan 60° = h/15
v3 = h/15
h = 15v3 ˜ 15 × 1.732 = 25.98m
Problem 2: The angle of elevation of the top of a tower from a point on the ground, which is 30m away from the foot of the tower, is 30°. Find the height of the tower.
Solution:
tan 30° = h/30
1/v3 = h/30
h = 30/v3 = 30v3/3 = 10v3 ˜ 17.32m
Problem 3: A kite is flying at a height of 60m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string.
Solution:
sin 60° = 60/Length of string
v3/2 = 60/L
L = 60 × 2/v3 = 120/v3 = 40v3 ˜ 69.28m
11. Special Case: Two Angles of Elevation
When angles of elevation from two points are given:
Let height = h, distances = d1 and d2
h = d1 tan ?1 = d2 tan ?2
If points are collinear: d2 = d1 + distance between points
12. Important Points to Remember
- Always draw diagram first
- Angle of elevation = Angle of depression (alternate angles)
- Use tan when height and distance are involved
- Use sin when height and hypotenuse are involved
- Use cos when distance and hypotenuse are involved
- Answers should include proper units (m, cm, km)
- Round off to reasonable decimal places